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Variational approaches for signal and image processing

One-day workshop
When Nov 18, 2022
from 10:00 to 05:00
Attendees Barbara Pascal / Audrey Repetti / Julian Tachella / Titouan Vayer
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Location: Salle 101, ENSL



10h00 - 10h30: Sai Kiran Kadambari (Indian Institute of Science, Bangalore)
Product graph learning with applications to spectral clustering.
10h30 - 11h30: Barbara Pascal (CNRS, École Centrale Nantes)
Proximal schemes for the estimation of the reproduction number of Covid19 pandemic:from convex optimization to Monte Carlo sampling.
11h30-12h30: Audrey Repetti (Heriot-Watt University)
Learning Maximally Monotone Operators for Image Recovery

14h-15h: Titouan Vayer (INRIA & LIP, ENSL)
Towards Compressive Recovery of Sparse Precision Matrices

15h-16h: Julian Tachella (CNRS & LP, ENSL)
Imaging with Equivariant Deep Learning
16h-16h30: Siddartha Reddy Thummaluru (Indian Institute of Science, Bangalore)
Sampling and reconstruction of Non-Bandlimited Diffusive Sources on Graphs



Abstract Kadambari Sai Kiran : This work focuses on learning product graphs from multi-domain data. We assume that the product graph is formed by the Cartesian product of two smaller graphs, which we refer to as graph factors. We pose the product graph learning problem as the problem of estimating the graph factor Laplacian matrices. To capture local interactions in data, we seek sparse graph factors and assume a smoothness model for data. We propose an efficient iterative solver for learning sparse product graphs from data. We then extend this solver to infer multi-component graph factors with applications to product graph clustering by imposing rank constraints on the graph Laplacian matrices. Although working with smaller graph factors is computationally more attractive, not all graphs readily admit an exact Cartesian product factorization. To this end, we propose efficient algorithms to approximate a graph by a nearest Cartesian product of two smaller graphs. The efficacy of the developed framework is demonstrated using several numerical experiments on synthetic and real data.

Abstract Barbara Pascal : Monitoring the Covid19 pandemic constitutes a critical societal stake that received considerable research efforts. Raw infection counts are not informative enough about the pandemic spread dynamics, and one has to recourse to more advanced epidemiological indicators, the most popular being the reproduction number, defined in Cori's model as the average number of secondary cases caused by an infected individual. Though, the quality of Covid19 data, consisting in daily new infection counts reported by public health authorities, is low (due, e.g., to pseudo-seasonalities, irrelevant or missing counts), making robust estimation of daily reproduction numbers very challenging.

A first approach consists in designing a nonsmooth convex functional,whose minimization performs jointly a correction of erroneous counts and a temporal regularization, yielding epidemiologically relevant piecewise linear estimates of the reproduction number. A second approach aims at enriching the aforementioned pointwise estimates with a level of confidence, which is crucial for sanitary policies design and assessment. The variational approach have thus been reinterpreted in a Bayesian framework permitting credibility interval estimation. Leveraging proximal schemes used for nonsmooth functional minimization, proximal Langevin-based Monte Carlo sampling algorithms have been derived. These Markov chain Monte Carlo methods yield credibility interval estimates of both the reproduction number and of corrected new infection counts for more than 200 countries worldwide. 
The purpose of this talk is to draw a path from variational formulations, designed in an inverse problem spirit, to novel Markov chain Monte Carlo methods, intertwining Langevin proposal mechanisms and nonsmooth convex optimization tools to yield Bayesian estimates. The developed tools will be illustrated at work on real Covid19 data.



Abstract Audrey Repetti : We introduce a new paradigm for solving regularized variational problems. These are typically formulated to address ill-posed inverse problems encountered in signal and image processing. The objective function is traditionally defined by adding a regularization function to a data fit term, which is subsequently minimized by using iterative optimization algorithms. Recently, several works have proposed to replace the operator related to the regularization by a more sophisticated denoiser. These approaches, known as plug-and-play (PnP) methods, have shown excellent performance. Although it has been noticed that, under some Lipschitz properties on the denoisers, the convergence of the resulting algorithm is guaranteed, little is known about characterizing the asymptotically delivered solution. In the current article, we propose to address this limitation. More specifically, instead of employing a functional regularization, we perform an operator regularization, where a maximally monotone operator (MMO) is learned in a supervised manner. This formulation is flexible as it allows the solution to be characterized through a broad range of variational inequalities, and it includes convex regularizations as special cases. From an algorithmic standpoint, the proposed approach consists in replacing the resolvent of the MMO by a neural network (NN). We present a universal approximation theorem proving that nonexpansive NNs are suitable models for the resolvent of a wide class of MMOs. The proposed approach thus provides a sound theoretical framework for analyzing the asymptotic behavior of first-order PnP algorithms. In addition, we propose a numerical strategy to train NNs corresponding to resolvents of MMOs. We apply our approach to image restoration problems and demonstrate its validity in terms of both convergence and quality.

Abstract Julian Tachella: From early image processing to modern computational imaging, successful models and algorithms have relied on a fundamental property of natural signals: symmetry. Here symmetry refers to the invariance property of signal sets to transformations such as translation, rotation or scaling. Symmetry can also be incorporated into deep neural networks in the form of equivariance, allowing for more data-efficient learning. While there has been important advances in the design of end-to-end equivariant networks for image classification in recent years, computational imaging introduces unique challenges for equivariant network solutions since we typically only observe the image through some noisy ill-conditioned forward operator that itself may not be equivariant. In this talk, I will present the emerging field of equivariant imaging and show how it can provide improved generalization and new imaging opportunities. Along the way I will show the interplay between the acquisition physics and group actions and links to iterative reconstruction, blind compressed sensing and self-supervised learning.

Abstract Titouan Vayer: We consider the problem of learning a graph modeling the statistical relations of the $d$ variables of a dataset with $n$ samples. Standard approaches amount to searching for a precision matrix $\boldsymbol\Theta$ representative of a Gaussian graphical model that adequately explains the data. However, most maximum likelihood based estimators usually require to store the $d^{2}$ values of the empirical covariance matrix, which is often too costly in a high-dimensional setting. In this work we adopt a ‘‘compressive'' viewpoint and look for estimating a sparse $\boldsymbol\Theta$ from a \emph{sketch} of the data, \textit{i.e.} a low-dimensional vector of size $m << d^{2}$ carefully designed from the data using nonlinear random features. Under certain assumptions on the spectrum of $\boldsymbol\Theta$, we show that it is possible to estimate it robustly from a sketch of size $m=\mathcal{O}\left((d+2k)\ln(d)\right)$ where $k$ is the maximal number of edges of the underlying graph. We also show that our estimator has an error that decreases in $\mathcal{O}(n^{-1/2})$. These guarantees are inspired from the compressed sensing theory and involve restricted isometric properties and instance optimal decoders. Our estimator requires solving a non-convex inverse problem and we investigate the possibility of achieving practical recovery by a variant of the Davis-Yin three operator splitting algorithm. We compare our approach with ‘‘Graphical LASSO'' type estimators on synthetic datasets. Finally, we discuss in a last part the limitations and perspectives of this work, which partially answers some questions but also opens many others.
Abstract Thummaluru Siddartha ReddyIn this talk, we discuss on sampling and reconstruction of non-bandlimited diffusive sources (diffusive and wave fields) modeled by heat and wave equations on graphs. More specifically, we discuss the techinques to recover the underlying sources that induces field (diffusive/wave) in the network by subsampling that essentially accounts to considering the measurements from the smaller subset of nodes. Further, we talk on considering two special scenarios on the field (diffusive/wave) induced in the network by the driven sources and discuss the simple least squares estimator for the recovery of sources. When the total field induced in the network is due to either of the initial distribution field or the external input, underlying sources can be recovered from observations without imposing any spectral constraints on the sources. When total field induced in the network is due to the presence of both initial distribution field and external input, underlying sources can be recovered exactly from observations by restricting one of the sources to be bandlimited. Further, to determine the nodes where measurements are to be collected,  we discuss a designed sparse sampler based on a greedy algorithm which maximizes the submodular cost function and results an optimal sampling set.  Several experiments are performed on both synthetic  real data to validate the theory will be discussed. More specifically, we discuss the developed theory on diffusive fields on localizing and recovering hot spots on a rectangular metal plate with a cavity, source localization on epidemic networks. Where as the developed theory on wave fields is applied for estimating the epicenter in seismic events.